Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-7x+y &= -3 \\ -3x-y &= -6\end{align*}$
Begin by moving the $x$ -term in the second equation to the right side of the equation. $-y = 3x-6$ Divide both sides by $-1$ to isolate $y$ $y = {-3x + 6}$ Substitute this expression for $y$ in the first equation. $-7x+({-3x + 6}) = -3$ $-7x - 3x + 6 = -3$ Simplify by combining terms, then solve for $x$ $-10x + 6 = -3$ $-10x = -9$ $x = \dfrac{9}{10}$ Substitute $\dfrac{9}{10}$ for $x$ back into the top equation. $-7( \dfrac{9}{10})+y = -3$ $-\dfrac{63}{10}+y = -3$ $y = \dfrac{33}{10}$ $y = \dfrac{33}{10}$ The solution is $\enspace x = \dfrac{9}{10}, \enspace y = \dfrac{33}{10}$.